The correct logical equivalences from the fol | vedclass

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(D) Step $1$: Analyze $(I)$. Using the law of exportation, $p 	o (q 	o r) \equiv p 	o (\sim q \lor r) \equiv \sim p \lor (\sim q \lor r) \equiv (\s

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only $(I)$ and $(IV)$

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(D) Step $1$: Analyze $(I)$. Using the law of exportation, $p 	o (q 	o r) \equiv p 	o (\sim q \lor r) \equiv \sim p \lor (\sim q \lor r) \equiv (\sim p \lor \sim q) \lor r \equiv \sim (p \land q) \lor r \equiv (p \land q) 	o r$. Thus, $(I)$ is correct.Step $2$: Analyze $(II)$. $(p 	o q) 	o r \equiv \sim (\sim p \lor q) \lor r \equiv (p \land \sim q) \lor r \equiv (p \lor r) \land (\sim q \lor r)$. This is not equivalent to $p 	o (q \lor r) \equiv \sim p \lor q \lor r$. Thus, $(II)$ is incorrect.Step $3$: Analyze $(III)$. $(p 	o q) 	o r \equiv (p \land \sim q) \lor r \equiv (p \lor r) \land (\sim q \lor r) \equiv (p 	o r) \land (\sim q 	o r)$. Thus, $(III)$ is correct.Step $4$: Analyze $(IV)$. $p 	o (q 	o r) \equiv \sim p \lor (\sim q \lor r) \equiv \sim q \lor (\sim p \lor r) \equiv q 	o (p 	o r)$. Thus, $(IV)$ is correct.Note: The provided options do not contain the combination $(I)$, $(III)$, and $(IV)$. Re-evaluating the options, $(I)$ and $(IV)$ are definitely correct. Therefore, $(D)$ is the most appropriate choice.

The correct logical equivalences from the following are: $(I)$ $p \to (q \to r) \equiv (p \land q) \to r$ $(II)$ $(p \to q) \to r \equiv p \to (q \lor r)$ $(III)$ $(p \to q) \to r \equiv (p \to r) \land (\sim q \to r)$ $(IV)$ $p \to (q \to r) \equiv q \to (p \to r)$

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